1. dielectric polarization 2. method of images 3...

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Lecture 7 Electrostatics (4)

1. Dielectric Polarization

2. Method of Images

3. Electrostatic Energy

4. Boundary Conditions

5. Coding Example

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Dielectric polarization- Negative/positive charges oriented toward the positive/negative electrode- Charges do not move freely in the material but reoritented.

1. Dielectric Materials and Permittivity

Fig. Polarization in materials [www.pcbdirectory.com]

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Polarization mechanisms in dielectric materials- Electron displacements- Ion displacements- Dipole orientation- Space (interfacial) charge displacements

Fig. Polarization mechanisms [Knowles]

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Polarization charge density- The effect of polarization is modelled using charge densities.- Polarization charges cannot move around. → Bound charges

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20

0 0

polarization dipole moment per unit volume

ˆ1 (single dipole)4

ˆ1 (distributed dipoles)4

1 1 ( )4 4

ˆ (polarized bound surface charge density)

V

S V

b

VR

V dVR

dV dVR R

P

p R

P R

P S P

P n

(polarized bound volume charge density)b P

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Electric flux density in dielectric materials

0

0

0

: bound charge density due to polarization

: freely moving charge (fee charge) density

(Gauss's law)

( )

(electric flux density)

b f

b

f

b f f

f

f

E P

E P

D E P

D

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Permittivity of dielectric materials

0 0 0 0

0

0

(1 )

: electric susceptibility

(1 ) (permittivity)

1 (dielectric constant or relative permittivity)

e e

e

e

r e

D E P E E E E

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Example: dielectric in a parallel-plate capacitor

0

0| | | | Electric flux density is increased in the dielectric.

E E

D D

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Example: dielectric in an external field

0

00

0

(continuity of electric flux; Gauss's law)

| | | |

D D

E E

E E

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2. Method of Images

1

2

1 1 2 2

1 2

1 23 31 2

( , , ) : field point

Original charge: at (0,0, )

Image charge: at (0,0, )

,

1 1( )4 4

ˆ ˆ1 14 4

q q

x y z

q d

q d

q qV V VR R

q qVR R

r

r

r

R r r R r r

r

R RE

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Example: a point source in a right-angle conductor

1

2

3

4

1 1 2 2 3 3 4

( , , ) : field point

at ( , ,0) : the original point charge

at ( , ,0) : an image charge



, , ,

x y z

q a b

q a b

q a b

q a b

r

r

r

r

r

R r r R r r R r r R r 4

1 2 3 4

31 2 43 3 3 31 2 3 4

1 1 1 1( )4

ˆˆ ˆ ˆ

4

qV rR R R R

qVR R R R

r

RR R RE

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Example: problems requiring an infinite number of images

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Example: a point charge outside a conducting sphere

1 1

2

aQ qL

aL

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1 1 1

11 2 2 2 2 2

2 1

1 1 21, 2 3 2 3 2 3

2 1 3 1 3 2

1 2 3, 2 3

1

1

only: 0 since no field exists before

, : 4 | |

, :4 | | 4 | | 4 | |

, , ..., : ...

14 | |

N N

ij

ii jj

q W q V

qq q W W q V q

q q qq q q W W W q q

q q q q W W W W

qq

r r

r r r r r r

r r1

1 1

1 1,2 4 | |

N

i

N Nj

i i ii ji j

j i

qq V V

r r

Energy stored in N point charges

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Energy stored in distributed charges

1

12

( )

1 1 12 2 2

N

i ii

i L s v

i

L s vL S V

W q V

q dL dS dV

V

W dL dS dV

r

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Electrostatic energy in terms of electric field

2

1 1( ) ( )2 2

( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1( ) ( ) ( )2 2 2 2

1 (surface density of electric energy, J/m )2

1 2

v vV V

V V S V

s

v

W VdV VdV

V V V V V V V

W V dV dV V d dV

w V

w

D D

D D D D D D D D E

D D E D S D E

D n

D E

3(volume density of electric energy, J/m )

1 0 as 2

1( )2

S

V

V d V

W dV

D S

D E

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Example: a charged sphere

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Boundary conditions on a dielectric interface

4. Boundary Conditions

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1 2

1 2

1 2

1 2

1 21 2

1 2

( )

0 ,

n n s

n n s

n ns

n ns n n

D dS D D S Q S

D D

E E

E ED D

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Boundary conditions on a dielectric-conductor interface

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Example: infinite charged plate

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Example: angles of electric field lines on a dielectric interface

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5. Coding Example

Example: angles of electric field lines on a dielectric interfaceWrite down a Python code for the stored energy for N point charges.Input data:N : number of point chargesqi (C), (xi, yi, zi) m : i-th point charge and positionOutput data:W (J): stored energy

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# Lecture 7 - Python coding example: Stored energy in n point charges# Input data:# n = number of point charges (maximum 100)# qi = charge (C) of the i-th charge# (xi,yi,zi) = position (m) of the i-th charge# Output data:# w = stored energy (J)

from math import *e0=8.853e-12;pi=3.141593x=[0]*100;y=[0]*100;z=[0]*100;q=[0]*100while True:

n=int(input('Number of point charges='))for i in range(n):print('For charge no.',i)q[i]=float(input('Charge: q(C)='))x[i],y[i],z[i]=map(float,input('Position: x,y,z(m)=').split())

w=0for i in range(n):sum=0for j in range(n):

if(j == i):continue

else:sum=sum+q[j]/sqrt((x[i]-x[j])**2+(y[i]-y[j])**2+(z[i]-z[j])**2)

w=w+q[i]*sumw=w/(4*pi*e0)/2print('Stored: W(J)=',w)

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'''Number of point charges=3For charge no. 0Charge: q(C)=1e-10Position: x,y,z(m)=1 1 3For charge no. 1Charge: q(C)=2e-4Position: x,y,z(m)=3 -1 1For charge no. 2Charge: q(C)=4e-10Position: x,y,z(m)=4 1 -4Stored: W(J)= 0.00018318585243Number of point charges='''

1. dielectric polarization 2. method of images 3...

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